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THE DOUBTER
it from slavery in Euclid's too narrow economy. And just as the
inventors of non-Euclidean geometry revealed vast and hitherto
unsuspected horizons to geometry and physical science, so the
creators of the theory of algebraic numbers uncovered an
entirely new light, illuminating the whole of arithmetic and
throwing the theories of equations, of systems of algebraic
curves and surfaces, and the very nature of number itself, into
sharp relief against a firm background of shiningly simple
postulates.
The creation of 'ideals* - Dedeldnd's inspiration from Rum-
mer's vision of 'ideal numbers' - renovated not only arithmetic
but the whole of the algebra which springs from the theory of
algebraic equations and systems of such equations, and it
proved also a reliable clue to the inner significance of the
*enumerative geometry' * of Pliicker3 Cayley and others, which
absorbed so large a fraction of the energies of the geometers of
the nineteenth century who busied themselves with the inter-
sections of nets of curves and surfaces. And last, if Kronecker's
heresy against Weierstrassian analysis (noted later) is some day
to become a stale orthodoxy, as all not utterly insane heresies-
sooner or later do, these renovations of our familiar integers,
1,2,3, ... , on which all analysis strives to base itself, may ulti-
mately indicate extensions of analysis, and the Pythagorean
speculation may envisage generative properties of 'number*
that Pythagoras never dreamed of in all his wild philosophy.
Kronecker entered this beautifully difficult field of algebraic
numbers in 1845 at the age of twenty-two with his famous
dissertation De Unitatibus Compleods (On Complex Units). The
particular units he discussed were those hi algebraic number
fields arising from the Gaussian problem of the division of the
cb/cumference of a circle into n equal axes. For this work he got
Ms Ph.D.
The German universities used to have - and may still have -
* One problem in this subject: an algebraic curve may have loops
on it, or places where the curve crosses its tangents; given the degree
of the curve, how many such points are there? Or if we cannot
answer that, what equations connecting Ubie number of these and
otaer exceptional points must hold? Similarly for surfaces.
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